Figure 4. Dual constraint rectify In calculation we use the distance between the selected points
and the seed point weighted the scale coefficient. The weigh function is same as multi-level B-spline interpolation we
described in section 2.2. Assume the number of the seed point pairs is n. The i-th seed point’s contribution to selected points is
ρ
i
, We calculate the scale coefficient as formula 9. s
= ∑ ρ
i n
i
s
i
9 Dual constraint rectify matrix is utilized to affine the matching
area. The final matching image blocks are shown as W
1
and W
2
in figure 4. In this way a lot of redundant computation can be reduced.
After dual constraint rectify, the NCC Normalized cross- correlation methods is employed for fine match. According to
the NCC similarity, a 3
×3 area centered by the matched point is selected to do a polynomial fitting. At last, the optimized
matching point is determined as the extreme point in the polynomial curve. The polynomial denote as follows.
fx ,y=a
x
2
+a
1
y
2
+a
2
xy+a
3
x+a
4
y+a
5
10 In formula 10, we denote fx
,y=1NCCx,y .Use Gaussian function to weigh each pixel’s contribution by distance. The
final result is generated by least square method. x
opt
= 2a
1
a
3
-a
2
a
4
a
2 2
-4a
1
a y
opt
= 2a a
4
-a
2
a
3
a
2 2
-4a
1
a
2.4 RANSAC based mismatching elimination
RANSAC Random Sample Consensus was first proposed by Fischer and Bolles in 1981. The basic assumption of RANSC is
the test sample involved with both inliers and outliers, the outliers are caused by noises, wrong assumptions or
miscalculations. If a correction test sample is given. The corresponding parameter model can be determined. The
RANSAC theory can be described as follows: Firstly, a mathematical model is selected. Assume that the
model can be determined by at least n parameters. An observation set P is given, contained m observations, mn.
Random select a sub set S from P. S involved with n observations. Obviously, S can determine a mathematical model.
Denote the mathematical model as M. Use all the observations in P to test model M. Set a test threshold. Passed observations
can build a new set S
1
. Denote S
1
as the model M’s consistent set. Second, compare the observation number of S
1
and the threshold t, if it larger than t. Use S
1
to determine the new mathematical model M
1
. Third, if the observations number of S
1
less than the threshold t. Use first step to find the S
1
can meet the demands that the observations number of S
1
no less than t. While the iteration time achieve a certain number and the result
still can’t meet the condition we set. RANSAC fails. In our approach, the test sample has a very large dataset. Use
the traditional RANSAC may have numerous iterations. Assume that the non-mismatching probability of the fine
matching result is p, the probability of acquiring at least n non- mismatching point pairs from a k times random selection in fine
matching results is Q. Then
1 − p
n �
denote the probability of the selected k subsets each one contained mismatching point
pairs. Thus k can be denoted as follows:
k =
ln1−Q ln1−p
n
11 From 11 we find that based on our assumption, RANSAC
iteration times are not affected by observations number. Normally the mismatching rate in PCA-SIFT is around 20. As
the proposed dense matching approach use the PCA-SIFT results as the initial input. The mismatching rate is related to
PCA-SIFT. Assume the mismatching rate of proposed approach is 30. Relative orientation needs at least 5 corresponding point
pairs. If we want the correct probability achieve 99. By calculation, k= 20.
In photogrammetry, the correspondence in a stereo pair fulfils the co-planarity condition. Thus, we use the co-planarity
condition as the mathematical model to operate RANSAC. Use least square method to calculate the error function. The point’s
residual larger than 3 times of the mean-error will be eliminated.
3. EXPERIMENTS AND ANALYSIS
